Typical subspace

Consider an ensemble ${\displaystyle \left\{p_{X}(x),\rho _{x}\right\}_{x\in {\mathcal {X}}}}$ of states. Suppose that each state ${\displaystyle \rho _{x}}$ has the following spectral decomposition:

${\displaystyle \rho _{x}=\sum _{y}p_{Y|X}(y|x)\vert y_{x}\rangle \langle y_{x}\vert .}$

Consider a density operator ${\displaystyle \rho _{x^{n}}}$ which is conditional on a classical sequence ${\displaystyle x^{n}\equiv x_{1}\cdots x_{n}}$:

${\displaystyle \rho _{x^{n}}\equiv \rho _{x_{1}}\otimes \cdots \otimes \rho _{x_{n}}.}$

We define the weak conditionally typical subspace as the span of vectors (conditional on the sequence ${\displaystyle x^{n}}$) such that the sample conditional entropy ${\displaystyle {\overline {H}}(y^{n}|x^{n})}$ of their classical labels is close to the true conditional entropy ${\displaystyle H(Y|X)}$ of the distribution ${\displaystyle p_{Y|X}(y|x)p_{X}(x)}$:

${\displaystyle T_{\delta }^{Y^{n}|x^{n}}\equiv {\text{span}}\left\{\left\vert y_{x^{n}}^{n}\right\rangle$ :\left\vert {\overline {H}}(y^{n}|x^{n})-H(Y|X)\right\vert \leq \delta \right\},}

where

${\displaystyle {\overline {H}}(y^{n}|x^{n})\equiv -{\frac {1}{n}}\log \left(p_{Y^{n}|X^{n}}(y^{n}|x^{n})\right),}$

${\displaystyle H(Y|X)\equiv -\sum _{x}p_{X}(x)\sum _{y}p_{Y|X}(y|x)\log p_{Y|X}(y|x).}$

The projector ${\displaystyle \Pi _{\rho _{x^{n}},\delta }}$ onto the weak conditionally typical subspace of ${\displaystyle \rho _{x^{n}}}$ is as follows:

${\displaystyle \Pi _{\rho _{x^{n}},\delta }\equiv \sum _{y^{n}\in T_{\delta }^{Y^{n}|x^{n}}}\vert y_{x^{n}}^{n}\rangle \langle y_{x^{n}}^{n}\vert ,}$

where we have again overloaded the symbol ${\displaystyle T_{\delta }^{Y^{n}|x^{n}}}$ to refer to the set of weak conditionally typical sequences:

${\displaystyle T_{\delta }^{Y^{n}|x^{n}}\equiv \left\{y^{n}:\left\vert {\overline {H}}\left(y^{n}|x^{n}\right)-H(Y|X)\right\vert \leq \delta \right\}.}$

The three important properties of the weak conditionally typical projector are as follows:

${\displaystyle \mathbb {E} _{X^{n}}\left\{{\text{Tr}}\left\{\Pi _{\rho _{X^{n}},\delta }\rho _{X^{n}}\right\}\right\}\geq 1-\epsilon ,}$

${\displaystyle {\text{Tr}}\left\{\Pi _{\rho _{x^{n}},\delta }\right\}\leq 2^{n\left[H(Y|X)+\delta \right]},}$

${\displaystyle 2^{-n\left[H(Y|X)+\delta \right]}\ \Pi _{\rho _{x^{n}},\delta }\leq \Pi _{\rho _{x^{n}},\delta }\ \rho _{x^{n}}\ \Pi _{\rho _{x^{n}},\delta }\leq 2^{-n\left[H(Y|X)-\delta \right]}\ \Pi _{\rho _{x^{n}},\delta },}$

where the first property holds for arbitrary ${\displaystyle \epsilon ,\delta >0}$ and sufficiently large ${\displaystyle n}$, and the expectation is with respect to the distribution ${\displaystyle p_{X^{n}}(x^{n})}$.

• Classical capacity
• Quantum information theory

• Wilde, Mark M., 2017, Quantum Information Theory, Cambridge University Press, Also available at eprint arXiv:1106.1145